Stabilization of Systems by Delayed Noisy States

In this article, stabilization of systems by delayed noisy states is investigated. The time delays in the stabilizing noisy states are extended into the general form. To support this novelty, the familiar Doob martingale inequality in the continuous version is improved; the equivalence principle, which says that the exponential stability in moment of an anhysteretic stochastic system infers the same property of the corresponding hysteretic system, is extended to the cases with the newly proposed semi-Lipschitz condition; the implication theorem, which indicates that the exponential stability in moment implies almost sure exponential stability, is generalized to the stochastic systems with time delays of general forms only under the linear growth condition. Technically, variable substitution is applied to time to map the function spaces for the functional differential equation models with unbounded time delays into those with bounded ones. Unified framework for the stability analysis for both bounded and unbounded time delays is built with the mapping, the decay rates, including those for polynomial stability, for estimates of the solutions with time delays of general form are formulated. At the end of the article, typical strategies, including the divided state feedback with unbounded time delays, for the stabilizing noise are presented, a numerical example is proposed to illustrate the method and to show the efficiency of the results of the paper.